1. Field of the Invention
This invention relates to graphical display of time series data, both from historical records and in real time.
2. Description of the Prior Art
When analyzing time series data, it is often important to evaluate not just the values of the data at different times, but also the fluctuations, that is, the changes from one observation time to the next. For example, in assessing the fair price of an option or a futures contract on a financial instrument, the most important determining factors are the current price of the underlying instrument, and the volatility of the underlying instrument. The volatility is the typical magnitude of price fluctuation from one time to the next.
A wide-spread practice in financial market analysis is to evaluate volatility numerically through the use of statistical models based on the concept of a random walk. These random walk models all depend on the assumption that the statistical distribution of price changes is a normal or log-normal distribution. However, the actual distribution of price changes in financial market data is known to be inconsistent with the assumption of normality or log-normality. In fact distributions of price changes in financial market data typically show strong leptokurtosis; that is, the distributions are strongly peaked near zero price change, and have fat tails, meaning that large price changes in either direction occur more often than they would in a normal or log-normal distribution. Alternative proposals for models based on other assumed forms of price change distribution have been made, but analysis of these alternative models is much more difficult and less complete than for the random walk models, and the alternative models have not been accepted and used in practice.
In such a situation where a widely used model is founded on a doubtful assumption, it would seem prudent to have some other means of assessing volatility which does not depend on any assumptions about the nature of the statistical distribution of price changes. Graphical display is one very effective method for comprehending numerical data directly, without reliance on modeling assumptions. However, the time-honored method of plotting time series data as a graph, with the data drawn as line segments or stair steps connecting data points, while effective for price data, is not so effective for representing price change data in a way that can be quickly and accurately comprehended. Furthermore, the conventional two-dimensional graph uses the display area in an inefficient manner: the information is contained in the lines or curves drawn on the graph, and the remaining blank area of the graph, both above and below the lines or curves, is needed to convey the information in the graph lines, but does not itself contain any information and is therefore is a sense wasted space. When it becomes necessary to evaluate rapidly a large number of time series e.g. of different financial instruments, this inefficient use of display area becomes a crippling disadvantage.
In "Recurrence Plots of Dynamical Systems" by Eckmann et al., Europhysics Letters, Vol. 4, No. 9, pp. 973-977 (1987), a method is disclosed for displaying recurrences, or close returns in phase space, of time series data using a two-dimensional image. Each axis of the image represents time, divided into discrete intervals. The image therefore consists of a grid of small squares, with each small square marked (black) or unmarked (white) depending on the values of the time series at the two corresponding times. A small square is marked (black) if the two values of the time series at the corresponding times are very close to each other; otherwise the small square is unmarked (white). (More generally, Eckmann et al. compare not just values at the two times, but rather they compare patterns of some specified length leading up to each of the two times; the comparison is made by treating each pattern as a vector in a multidimensional phase space, and computing the distance separating the two vectors.) The resulting two-dimensional image is called a recurrence plot.
Variants of this idea have also been disclosed (for example on the World Wide Web at http://www.scri.fsu.edu/.about.nayak/chaos/recur.html) using a plurality of colors to mark the small squares, with the colors chosen to represent the distance separating the two time series values or patterns. In accord with the specific goal of finding recurrences in time series data, colors are assigned to a range of distances which are all considered close, whereas distances not considered close always result in an unmarked square. Hot or bright colors represent very close recurrences, while cool or dark colors stand for recurrences which are close but not very close.
Several modifications or applications of recurrence plots have been disclosed, e.g. in "Topological analysis and synthesis of chaotic time series" by Mindlin and Gilmore, Physica D, Vol. 58, pp. 229-242 (1992); "Embeddings and delays as derived from quantification of recurrence plots" by Zbilut and Webber, Physics Letters A, Vol. 171, pp. 199-203 (1992); "A new approach to testing for chaos, with applications in finance and economics" by Gilmore, International Journal of Bifurcation and Chaos, Vol. 3, No. 3, pp. 583-587 (1993); "Dynamical assessment of physiological systems and states using recurrence plot strategies" by Webber and Zbilut, American Journal of Physiology, Vol. ?, pp. 965-973 (1994); "Properties of AE data and bicolored noise" by Takalo et al., Journal of Geophysical Research, Vol. 99, No. A7, pp. 13,239-13,249 (1994). In every instance the teaching specifies that only close distances should be marked, and all distances which are not close should remain unmarked.
The recurrence plot is a very efficient display method, conveying a great deal of information in the display surface area utilized. There is still some inefficiency resulting from the fact that all pairs of times having widely separated distances result in unmarked squares; these convey less information than is found in the colored squares, which distinguish among a plurality of ranges of small distances. However, this is entirely appropriate to the objective of identifying close recurrences in time series data.
Recurrence plots are very useful for identifying recurrences in time series data. However, the application of this display method to evaluating volatility has not been practiced. Indeed, as taught, the method would constitute an ineffective means for evaluating volatility, since it would only discriminate among the smaller price changes (or smaller differences in price changes), while moderate, large, and very large price changes (or differences in price changes) would all be lumped into one category, resulting in unmarked squares.
What is needed is a graphical display means that presents time series data in a form that permits rapid and accurate evaluation of changes in the data from one time to the next, discriminating clearly and strongly among small, moderate, and large price changes, while making efficient use of display surface area so that a plurality of time series can be visualized simultaneously within a limited total display area, such as a computer monitor.